The single copy of the Kerr solution is the gauge field [1]}[1]} claims a slightly different gauge
field since it sets \(M = 0\)  in order to take the flat background limit. Even though the \(dr\)  term is pure
gauge anyway, it is useful to transform to coordinates \((\mathring{t}, r, \theta ,\mathring{\phi })\) , whose relation to the Kerr-Schild coordinates \((x_{k},\mathring{\psi }_{j})\)  defined in equation (4.77) of
[3]} parallels the transformation to Boyer-Lindquist coordinates from canonical
coordinates (REF ). Then, we can
view \(\mathbf {A}\)  as living on the flat background with metric
\(\mathbf {\mathring{g}}\) .
Written in these coordinates, the gauge potential takes precisely the form in [1]} with no parameter
limits post hoc.
\( \mathbf {A} = \frac{Qr}{r^{2} + x_{1}^{2}} \left(dt + \frac{r^{2}+x_{1}^{2}}{r^{2} + a^{2} - 2Mr}\ dr + a\left( 1- \frac{x_{1}^{2}}{a^{2}}\right) d\phi \right).\) 
